If two parallel lines are intersected by a transversal, then alternate interior angles are congruent

So the question is, if we have two lines that might be parallel and they're intersected by a transversal, can we do the converse of the parallel lines theorem? Which says, if we have alternate interior angles or alternate exterior angles, or corresponding angles that are congruent, is that enough to say that these two lines are parallel? And as we read right here, yes it is. If two lines and a transversal form alternate interior angles, notice I abbreviated it, so if these alternate interior angles are congruent, that is enough to say that these two lines must be parallel.If you have alternate exterior angles. That is these two angles right here that are alternate exterior, if those two are congruent, you don't even need to know about these interior ones. That's enough to say that they're parallel.And finally, corresponding angles. If you have one pair of corresponding angles that are congruent you can say these two lines must be parallel. So the converse of the parallel lines there is true.

In Geometry, when any two parallel lines are cut by a transversal, many pairs of angles are formed. There is a relationship that exists between these pairs of angles. While some of them are congruent, the others are supplementary. Let us learn more about the angles formed when parallel lines are cut by a transversal.

What are Parallel Lines Cut by Transversal?

Parallel lines are straight equidistant lines that lie on the same plane and never meet each other. When any two parallel lines are intersected by a line (known as the transversal), the angles that are subsequently formed, have a relationship. The various pairs of angles that are formed on this intersection are Corresponding angles, Alternate Interior Angles, Alternate Exterior Angles and Consecutive Interior Angles. Observe the figure given below which shows two parallel lines 'a' and 'b' cut by a transversal 'l'.

Angles Formed by Parallel Lines Cut by Transversal

When parallel lines are cut by a transversal, four types of angles are formed. Observe the following figure to identify the different pairs of angles and their relationship. The figure shows two parallel lines 'a' and 'b' which are cut by a transversal 'l'.

Corresponding angles

When two parallel lines are intersected by a transversal, the corresponding angles have the same relative position. In the figure given above, the corresponding angles formed by the intersection of the transversal are:

• ∠1 and ∠5
• ∠2 and ∠6
• ∠3 and ∠7
• ∠4 and ∠8

It should be noted that the pair of corresponding angles are equal in measure, that is, ∠1 = ∠5, ∠2 = ∠6, ∠3 = ∠7, and ∠4 = ∠8

Alternate Interior Angles

Alternate interior angles are formed on the inside of two parallel lines which are intersected by a transversal. In the figure given above, there are two pairs of alternate interior angles.

It should be noted that the pair of alternate interior angles are equal in measure, that is, ∠3 = ∠6, and ∠4 = ∠5

Alternate Exterior Angles

When two parallel lines are cut by a transversal, the pairs of angles formed on either side of the transversal are named as alternate exterior angles. In the figure given above, there are two pairs of alternate exterior angles.

It should be noted that the pair of alternate exterior angles are equal in measure, that is, ∠1 = ∠8, and ∠2 = ∠7

Consecutive Interior Angles

When two parallel lines are cut by a transversal, the pairs of angles formed on the inside of one side of the transversal are called consecutive interior angles or co-interior angles. In the given figure, there are two pairs of consecutive interior angles.

It should be noted that unlike the other pairs given above, the pair of consecutive interior angles are supplementary, that is, ∠4 + ∠6 = 180°, and ∠3 + ∠5 = 180°.

Properties of Parallel Lines Cut by Transversal

When any two parallel lines are cut by a transversal they acquire some properties. In other words, any two lines can be termed as parallel lines if the following conditions related to the angles are fulfilled.

• Any two lines that are intersected by a transversal are said to be parallel if the corresponding angles are equal.
• Any two lines that are intersected by a transversal are said to be parallel if the alternate interior angles are equal.
• Any two lines that are intersected by a transversal are said to be parallel if the alternate exterior angles are equal.
• Any two lines that are intersected by a transversal are said to be parallel if the consecutive interior angles are supplementary.

Related Links

Check out the following links related to Parallel Lines Cut by Transversal.

• Parallel Lines
• Transversal

1. Example 1: Identify the corresponding angles in the figure which shows two parallel lines 'm' and 'n' cut by a transversal 't'.

   

   Solution: In the given figure, two parallel lines are cut by a transversal, and the corresponding angles in the figure are ∠1 and ∠3; and ∠2 and ∠5.

2. Example 2: Find the value of x in the given parallel lines 'a' and 'b', cut by a transversal 't'.

   

   Solution: The given parallel lines are cut by a transversal, therefore, the marked angles in the figure are the alternate interior angles which are equal in measure. This means, 8x - 4 = 60°, and 8x = 64, x = 8.

   Therefore, the value of x = 8.

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FAQs on Parallel Lines Cut by Transversal

Parallel lines are straight equidistant lines that lie on the same plane and never meet each other. A transversal is any line that intersects two straight lines at distinct points. When any two parallel lines are intersected by a transversal, various angles are formed. There is a relationship that exists between these pairs of angles.

What happens When Parallel Lines are Cut by a Transversal?

When any two parallel lines are cut by a transversal, there are various pairs of angles that are formed. These angles are corresponding angles, alternate interior angles, alternate exterior angles, and consecutive interior angles.

What are the Special Pairs of Angles Formed when Parallel Lines Cut by Transversal?

When parallel lines are cut by a transversal, there are 4 special types of angles that are formed - corresponding angles, alternate interior angles, alternate exterior angles, and consecutive interior angles. While the pairs of corresponding angles, alternate interior angles, alternate exterior angles are congruent, the pairs of consecutive interior angles are supplementary.

How to Calculate Angle Measures in Parallel Lines Cut by a Transversal?

The unknown angles can be easily calculated when two parallel lines are cut by a transversal. The following facts help in finding the unknown angles. When parallel lines are cut by a transversal,

When Two Parallel Lines are Cut by a Transversal, are the Corresponding Angles Congruent?

Yes, when two parallel lines are intersected by a transversal, the corresponding angles that are formed are congruent.

When Two Parallel Lines are Cut by a Transversal, are the Alternate Interior Angles Congruent?

Yes, when two parallel lines are intersected by a transversal, the alternate interior angles are congruent.

Alternate Interior Angles theorem states, if two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent. Alternate interior angles are the angles formed on the opposite sides of the transversal. The alternate interior angles can prove whether the given lines are parallel or not. Let us learn more about the alternate interior angles theorem, the proof, and solve a few examples.

What is the Alternate Interior Angles Theorem?

The alternate interior angles theorem states that if a transversal crosses the set of parallel lines, the alternate interior angles are congruent. In the figure given below, a set of parallel lines m and n are intersected by the transversal and the following pairs of alternate interior angles are formed: ∠1 and ∠2, ∠3 and ∠4.

Since the given lines m and n are parallel, therefore the alternate interior angles will be congruent. ∠1 = ∠2 and ∠3 = ∠4.

Interior angles on the same side of the transversal are called consecutive interior angles or co-interior angles in short. Co-interior angles are supplementary when the lines are parallel.

∠2 + ∠3 = 180° and ∠6 +∠7 = 180°

Corresponding angles are a pair of angles on the similar corners of each of two lines on the same side of the transversal line. Corresponding angles formed by two parallel lines and a transversal are equal. Corresponding angles in the above image are: ∠2 and ∠4, ∠1 and ∠3, ∠5 and ∠7, and ∠6 and ∠8.

Definition of Alternate Interior Angles

When two parallel lines are crossed by a transversal, the pair of angles formed on the inner side of the parallel lines, but on the opposite sides of the transversal are called alternate interior angles. Alternate angles are of two types: Alternate interior angles and Alternate exterior angles

• Alternate interior angles: Alternate interior angles are the angles formed when a transversal intersects two coplanar lines. They are on the inner side of the coplanar lines but are on the alternate opposite sides of the transversal.
• Alternate exterior angles: Alternate exterior angles are formed on the exterior of the coplanar lines but on the alternate opposite sides of the transversal.

According to the image below:

Alternate interior angles are: ∠3 and ∠6, ∠4 and ∠5.

Alternate exterior angles are: ∠1 and ∠8, ∠2 and ∠7.

For example: Let us try to spot alternate interior angles in the given figure. Remember the lines do not have to be always parallel for alternate angles to be formed.

Alternate interior angles are: ∠3 and ∠6, ∠4 and ∠5

Alternate exterior angles are: ∠1 and ∠8, ∠2 and ∠7

Properties of Alternate Interior Angles

Here are a few properties of the alternate interior angles:

• The angles are congruent.
• Two lines that never intersect, are equidistant, and are coplanar are called parallel lines. The symbol for parallel to is II.
• If we have two lines (they don't have to be parallel) and have a third line that crosses them, then the crossing line is said to be a transversal.
• The sum of the angles formed on the same side of the transversal which are inside the two parallel lines is always equal to 180°.
• In the case of non – parallel lines, alternate interior angles don’t have any specific properties.

Finding Alternate Interior Angles

For finding alternate interior angles, we use the Z test. Make a zig-zag line including the parallel lines as shown in the diagram. Here, p and q are alternate interior angles. Similarly, a and d are also alternate interior angles.

Alternate Interior Angles Theorem and Proof

Statement: The theorem states that if a transversal intersects parallel lines, the alternate interior angles are congruent.

Given: Line p II line q

To prove: ∠2= ∠7 and ∠3 = ∠6

Proof: Suppose p and q are two parallel lines and t is the transversal that intersects p and q.

We know that, if a transversal intersects any two parallel lines, the corresponding angles and vertically opposite angles are congruent.

Therefore,

∠1 = ∠3 ………..(i) [Corresponding angles]

∠1 = ∠6 ………..(ii) [Vertically opposite angles]

From equations (i) and (ii), we get-

∠3 = ∠6 ..................[Alternate interior angles]

Similarly,

∠2 = ∠7

Hence, it is proved.

Alternate Interior Angles Theorem Converse

The converse of alternate interior angles theorem states that if two lines are intersected by a transversal forming congruent alternate interior angles, then the lines are parallel. Thus according to the converse of alternate interior angles theorem, the below-given lines will be parallel if ∠D is 40° and ∠B is 140°. This is because their corresponding alternate interior angles are of the measure 40° and 140°.

In conclusion, the alternate interior angles theorem states that the alternate interior angles will be equal if the lines are parallel, whereas its converse states that lines will be parallel if the alternate interior angles are congruent.

Co-interior Angles Theorem and Proof

Co-interior angles are the two interior angles that are on the same side of the transversal which makes it supplementary and sums up to 180 degrees. The co-interior angles resemble the shape C due to the placement of the angles but the angles are not equal to each other. The other names for co-interior angles are consecutive interior angles or the same side interior angles. The image below shows the shape of co-interiors.

Theorem: If the transversal intersects the two parallel lines, each pair of co-interior angles sums up to 180 degrees (supplementary angles).

Proof:

Let us consider the image given below:

In the figure, angles 3 and 5 are the co interior angles and angles 4 and 6 are the co-interior angles.

To prove: ∠3 and ∠5 are supplementary and ∠4 and ∠6 are supplementary.

Given that, a and b are parallel to each other and t is the transversal.

By the definition of linear pair,

∠1 and ∠3 form the linear pair.

Similarly, ∠2 and ∠4 form the linear pair.

By using the supplement postulate,

∠1 and ∠3 are supplementary

(i.e.) ∠1 + ∠3 = 180

Similarly,

∠2 and ∠4 are supplementary

(i.e.) ∠2 + ∠4 = 180

By using the corresponding angles theorem, we can write

∠1 ≅∠5 and ∠2 ≅ ∠6

Thus, by using the substitution property, we can say,

∠3 and ∠5 are supplementary and ∠4 and ∠6 are supplementary.

Hence, the co-interior angle theorem (consecutive interior angle) is proved.

The converse of this theorem is if a transversal intersects two lines, such that the pair of co-interior angles are supplementary, then the two lines are parallel.

Related Topics

Listed below are a few interesting topics related to the alternate interior angles theorem, take a look.

• Vertical Angles
• Alternate Angles
• Same Side Interior Angles

1. Example 1: Cathy has been asked to find the pairs of alternate interior angles from the given diagram. Can you help her?

   

   Solution: The alternate interior angles lie on "alternate" sides of the transversal, and they must be in the "interior" of the two parallel lines. The alternate interior angles are: ∠q and ∠z, ∠r and ∠a.

2. Example 2: Julia asks her students if they can find the relationship between \(\angle q\) and \(\angle d\) if the given lines \(m\) and \(n\) are parallel. Can you tell how are \(\angle q\) and \(\angle d\) related to each other?

   

   Solution: The given lines are parallel and are intersected by a transversal.

   According to the alternate interior angles theorem, the two congruent alternate interior angles are produced. They are: ∠q and ∠d,∠c and ∠p.

   Therefore, ∠q and ∠d are congruent alternate interior angles.

3. Example 3: If the given two lines are parallel and are intersected by a transversal, what will be the measure of ∠X and ∠Y?

   

   Solution: The given lines are parallel and according to the alternate interior angles theorem, the given angle of measure 85° and ∠X are congruent alternate interior angles.

   Therefore, 85° = ∠X
   Similarly, 95° and Y are congruent alternate interior angles. Therefore, 95° = ∠Y.

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FAQs on Alternate Interior Angles Theorem

The alternate interior angles theorem states that if a transversal crosses the set of parallel lines, the alternate interior angles are congruent. Two lines that never intersect, are equidistant, and are coplanar are called parallel lines. The symbol for parallel to is II. If we have two lines (they don't have to be parallel) and have a third line that crosses them, then the crossing line is said to be a transversal.

What is the Converse of the Alternate Interior Angles Theorem?

According to the converse of the alternate interior angles theorem, if a transversal intersects two lines such that the alternate interior angles are equal, then the two lines are said to be parallel.

Are Alternate Interior Angles Congruent?

Yes, alternate interior angles are congruent.

How is the Alternate Interior Angles Theorem and the Alternate Exterior Angles Theorem Alike?

The alternate interior angles theorem states that if two parallel lines are cut by a transversal, then the alternate interior angles are congruent. The alternate exterior angles theorem states that if two parallel lines are cut by a transversal, then the alternate exterior angles are congruent. Thus, the two theorems are alike with respect to the congruent angles produced.